Relation of the angular momentum of surface modes to the position of their power-flow center

We show that the value of the total angular momentum (AM) carried by a surface mode can be interpreted as representing the transverse position of the center or balance point of the power flow through the mode. Especially in the lossless cases, the value of the Abraham AM per unit power (multiplied by the square of the speed of light in vacuum) is exactly the same as the transverse position of this power-flow center. However, the Minkowski counterpart becomes proportional to that position with a coefficient in the form of 1 + η , where η is determined mainly by the constitutive parameters of media. © 2014 Optical Society of America OCIS codes: (160.3918) Metamaterials; (230.7370) Waveguides; (240.6690) Surface waves. References and links 1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). 2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). 3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). 4. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007). 5. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photon. 2, 351–354 (2008). 6. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007). 7. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of threedimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011). 8. K.-Y. Kim, J. Kim, I.-M. Lee, and B. Lee, “Analysis of transverse power flow via surface modes in metamaterial waveguides,” Phys. Rev. A 85, 023840 (2012). 9. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85, 061801(R) (2012). 10. K.-Y. Kim, I.-M. Lee, J. Kim, J. Jung, and B. Lee, “Time reversal and the spin angular momentum of transverseelectric and transverse-magnetic surface modes,” Phys. Rev. A 86, 063805 (2012). 11. A. Canaguier-Durand, A. Cuche, C. Genet, and T. W. Ebbesen, “Force and torque on an electric dipole by spinning light fields,” Phys. Rev. A 88, 033831 (2013). 12. X. Xiao, M. Faryad, and A. Lakhtakia, “Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. Part V: Spin and orbital angular momentums,” J. Nanophoton. 7, 073081 (2013). 13. K.-Y. Kim, “Origin of the Abraham spin angular momentum of surface modes,” Opt. Lett. 39, 682–684 (2014). 14. K.-Y. Kim, “Transverse spin angular momentum of Airy beams,” IEEE Photon. J. 4, 2333–2339 (2012). 15. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008). 16. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). #225148 $15.00 USD Received 16 Oct 2014; revised 18 Nov 2014; accepted 19 Nov 2014; published 25 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030184 | OPTICS EXPRESS 30184 17. K.-Y. Kim, I.-M. Lee, and B. Lee, “Grating-induced dual mode couplings in the negative-index slab waveguide,” IEEE Photon. Technol. Lett. 21, 1502–1504 (2009). 18. K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express 19, 2286–2293 (2011). 19. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Constraining validity of the Minkowski energy-momentum tensor,” Phys. Rev. A 79, 023813 (2009). 20. S. M. Barnett, “Resolution of the Abraham–Minkowski dilemma,” Phys. Rev. Lett. 104, 070401 (2010). 21. B. A. Kemp, “Resolution of the Abraham–Minkowski debate: implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109, 111101 (2011). 22. W. Wu, E. Kim, E. Ponizovskaya, Y. Liu, Z. Yu, N. Fang, Y. R. Shen, A. M. Bratkovsky, W. Tong, C. Sun, X. Zhang, S.-Y. Wang, and R. S. Williams, “Optical metamaterials at near and mid-IR range fabricated by nanoimprint lithography,” Appl. Phys. A 87, 143–150 (2007).


Introduction
Metamaterials have captured the interest of many researchers in optics and photonics fields because the real parts of their complex constitutive parameters (permittivity ε and permeability µ) and/or refractive indexes can be negative.This unique property has brought out groundbreaking applications such as superlenses [1], invisibility cloaks [2,3], 'trapped rainbow' storage of light [4], and so-called lasing 'spasers' [5] (for a comprehensive review, see [6,7]).Metamaterials have also been studied extensively in the context of optical waveguides because near the interface between two media having opposite signs of constitutive parameters, there can exist a new class of waveguide mode, a surface mode.
Usually, surface modes are thought to propagate only in their longitudinal directions, i.e., along the medium interfaces.However, as was shown in [8], the actual power guided through a surface mode is transported in a vortex-like way (see Fig. 1).That is, some of the optical power originally moving forward in one layer comprising the interface is transported transversely to the other layer and made to flow longitudinally again in a backward direction.This feature makes us investigate the angular momentum (AM) of light propagating in a form of surface mode.
Recently, several groups have reported their studies on the spin AM of surface modes [9-13] or similar (1+1)D waves [14].As is well known, the spin AM of an optical beam induces the rotation of particles around their own axes which are parallel to the propagating direction of the beam [15,16].However, the spin AM of a surface mode is a transverse one, i.e., its direction is normal to the propagating axis of the mode.In [13], one of the present authors proved that this transverse spin AM results from the rotation of the electric field component of the surface mode.In this paper, we extend our study to the (total) AM and suggest a way of interpreting it as a quantity related to the transverse position of the center or balance point of the power flow through the surface mode.As will be shown later, this interpretation enables us to estimate the qualitative features of AM quite easily.In addition, the differences between the Abraham and Minkowski versions of AM are identified in the proportional coefficients between the AM and the position of the power-flow center.

Mathematical expressions
Let us consider a material interface (with a normal vector ẑ), which supports a surface mode along the +x direction [see Fig. 1(a)].In the case of TM polarization, the magnetic field of this mode can be written as h = ŷ[φ r (z) cos(β r x − ωt)− φ i (z) sin(β r x − ωt)] exp(−β i x), where ω is the angular frequency of light in vacuum and φ (z) (= φ r + iφ i ) denotes the complex transverse profile of the surface mode whose propagation constant has been written as β = β r + iβ i .β r becomes positive and negative in forward and backward modes, respectively [17,18], while β i is always nonnegative.The above mode field becomes that of the electric component (e) if we assume TE-polarized configuration.Following [8], we can write the (time-averaged) linear  momentum density carried by this mode as where the prime indicates differentiation with respect to z.
For the past hundred years, extensive efforts have gone into the resolution of the so-called Abraham-Minkowski debate on the exact form of the linear momentum of light in a medium [19][20][21].From the Abraham's point of view, the momentum of light becomes inversely proportional to the refractive index of the medium, while Minkowski argued that it should be proportional to the refractive index.With the electromagnetic quantities, their forms of the linear momentum density are respectively given by g A = e × h/c 2 and g M = d × b, where d and b are the electric displacement and the magnetic flux density, respectively.We have not only theories but also experimental evidences in favor of either of them (see [10] and the references therein).In Eq. ( 1), this difference between g A and g M appears in the actual form of ξ (= ξ r + iξ i ).With the Abraham form, ξ is written as ξ A T E = µ/(|µ| 2 c 2 ) and ξ A T M = ε/(|ε| 2 c 2 ) for TE and TM modes, respectively (c is the speed of light in vacuum), while it becomes ξ M T E = ε and ξ M T M = µ when we take the Minkowski form.
The AM density j with respect to a reference point r 0 = (x 0 , z 0 ) is then given by (r − r 0 ) × g, resulting in The AM per unit length along the propagation direction (hereafter will be referred to as simply AM) can be defined by where Γ results from the waveguide loss, its integrand vanishing in the lossless medium.If we put the relative permittivity and relative permeability of the medium as ε r and µ r , the complex decay constant (κ = κ r + iκ i ) can be written as κ 2 = β 2 − ε r µ r (ω/c) 2 .Writing the mode profile φ (z) as φ (z < 0) = exp(κ 1 z) and φ (z ≥ 0) = exp(−κ 2 z) (hereafter the subscript 1 and 2 indicate respectively the left and right layers) and after some manipulation, we can obtain (5)

Abraham AM
Let us look into the Abraham form specifically.In this case, we have g A = s/c 2 , where s denotes the Poynting vector [10].Then, Eq. ( 3) can be rewritten as By normalizing Eq. ( 6), we can obtain the Abraham AM of the surface mode per unit power where Φ is a constant expressing the direction of overall power flow S = s dz, given by tan Φ = Γ In Eq. ( 7), we can find that the first term on its right side is the expectation value z − z 0 with respect to the distribution of optical power flow.It thus represents the position of the center or balance point of the power flow.It is independent of the longitudinal coordinate x as can be easily seen in Eq. (3) (the exponential decay term is canceled out by the normalization).The second term of Eq. ( 7) actually amounts to the transversal shift the overall power flow gains during the propagation distance of (x − x 0 ) along the longitudinal axis [see Fig. 2(a)].However, it should be remembered that although the overall power flows obliquely, the (transverse) position of the power-flow center is conserved (this is quite evident from the translational symmetry of the waveguide).Therefore, the right-side terms of Eq. ( 7) correspond to the transverse coordinate of a point at the reference longitudinal coordinate x 0 from which the overall power flows obliquely and reaches the point at (x, z − z 0 ).This coordinate is what the Abraham AM of a unit-power surface mode at a longitudinal coordinate x stands for (multiplied by c 2 ), although its direction is not along the z axis but along the y axis.Let us apply this geometrical interpretation to actual examples.We considered two surface modes guided through interfaces between a negative-index metamaterial (NIM; z < 0) and silica (z > 0) and between ε-negative (ENG; z < 0) and µ-negative (MNG; z > 0) media.In both cases, the constitutive parameters of adopted materials (see the caption) are such that κ r 2 is smaller than κ r 1 , and thus the major power flow occurs in the right layer as in Fig. 1.We note that the selected values of ε r and µ r remain within the range that can be actually implemented using current technologies (see [22] for an example).7).It is notable that the direction of S(x), i.e, Φ remains constant even though its magnitude is dependent on x in lossy waveguides.Please note that Φ < 0 in this figure.(b) Abraham AMs of surface modes per unit power at the wavelength λ = 1550 nm.L p denotes the propagation length of the surface mode.Solid and dotted lines correspond, respectively, to the mode at the NIM (ε r = −2.2+ 0.5i and µ r = −0.8)-silicainterface and the mode at the ENG (ε r = −2 + 0.5i and µ r = 1)-MNG (ε r = 1.2 and µ r = −0.5 + 0.1i) interface.We took r 0 = (0, 0) where x = 0 denotes the launching position of light to the waveguide (see the inset).
In the implicit assumption of forward power transport, the major forward power can flow either in the left or right layer.When it flows in the right layer, resulting in a minor backward power flow in the left layer, we can easily discern that the power-flow center lies in the right layer or z > 0. This enables us to identify that J A 0 (0) with respect to r 0 = (0, 0) is positive, i.e., its direction is along the +y direction.Similarly, when the major forward power flows in the left layer, we have z < 0 and thus J A 0 (0) becomes along the −y direction.In our two examples, since the major power flows occur in the right layers, we can presume that J A 0 (0) is positive in both cases.We can confirm this in Fig. 2(b) in which we plotted the numerical values of Abraham AMs per unit power carried by respective surface modes.
Then, the remaining detail is whether J A 0 (x) increases or decreases during propagation.It is dependent on the sign of Φ or the transverse direction of S. To estimate this, let us compare Figs.1(b) and 1(c), especially the two filled (blue) arrows which represent the instantaneous power flow along the transverse direction.If the medium is lossless, then the amounts of these two power flows are the same.Therefore, if we time-average them, the overall power flow along the transverse direction vanishes, and Γ (which is proportional to s z dz) and Φ become zero as well.That is, J A 0 (x) remains constant during propagation.When the medium is lossy, however, the amounts of those two power flows become different: that of Fig. 1(b) gets larger than that of Fig. 1(c) due to the propagation loss.Therefore, in our examples, the overall power flow is tilted along the −z direction.We thus have Φ < 0 and J A 0 (x) increases during propagation.This feature can also be found in Fig. 2(b).

Minkowski AM
In the case of the Minkowski form, the situation becomes somewhat complex.For the simple physical intuition, let us restrict our discussion to the lossless cases where g M = (ε r µ r )s/c 2 [10].Then, the Minkowski AM of the surface mode per unit power is given by If we set sx = (ε r µ r − 1)s x , Eq. ( 9) becomes where • • • sx denotes the expectation value with respect to sx instead of s x .If we put z− z 0 sx = γ z − z 0 , Eq. ( 10) can be simply written as where η = γ( ε r µ r − 1).Equation (11) clearly shows that the Minkowski AM of a unit-power surface mode (multiplied by c 2 ) is proportional to the position of the power-flow center.The proportional coefficient is of the form of 1 + η where η is determined by the constitutive parameters of the media.This characteristic reminds us of the usual understanding that the Minkowski momentum includes both contributions of light and matter while only that of the field comprises the Abraham momentum [13,21].If we note that z − z 0 is determined by the mode profile, we can regard it as representing the AM contained in the light field.Only this comprises the Abraham AM.However, in the Minkowski counterpart, we can find additional term which is proportional to η.If ε r µ r = 1, i.e., the media are effectively free space, η vanishes and the AM to the Abraham AM.This fact indicates that the additional η-proportional term in Eq. ( 11) is the momentum contribution of the media.
To see the effect of the η-related term more clearly, we reconsidered the surface modes in Fig. 2(b) (however, neglecting material losses) and compared their Abraham and Minkowski AMs per unit power in Fig. 3.When the media comprising the interface are an NIM and silica, since ε r µ r > 1 everywhere in this example, we have ε r µ r − 1 > 0 and thus the Minkowski AM becomes larger than the Abraham AM.However, interestingly, when the

#Fig. 2 .
Fig. 2. (a) Geometrical interpretation of Eq.(7).It is notable that the direction of S(x), i.e, Φ remains constant even though its magnitude is dependent on x in lossy waveguides.Please note that Φ < 0 in this figure.(b) Abraham AMs of surface modes per unit power at the wavelength λ = 1550 nm.L p denotes the propagation length of the surface mode.Solid and dotted lines correspond, respectively, to the mode at the NIM (ε r = −2.2+ 0.5i and µ r = −0.8)-silicainterface and the mode at the ENG (ε r = −2 + 0.5i and µ r = 1)-MNG

#Fig. 3 .
Fig.3.Abraham and Minkowski AMs of surface modes per unit power.The same interfaces as those in Fig.2(b) were adopted neglecting material losses.Solid lines denote the Minkowski AMs while dotted lines show the Abraham counterparts.We also took r 0 = (0, 0).